Signature of quantum interference effect in inter-layer Coulomb drag in graphene-based electronic double-layer systems

The distinguishing feature of a quantum system is interference arising from the wave mechanical nature of particles which is clearly central to macroscopic electronic properties. Here, we report the signature of quantum interference effect in inter-layer transport process. Via systematic magneto-drag experiments on graphene-based electronic double-layer systems, we observe low-field correction to the Coulomb-scattering-dominated inter-layer drag resistance in a wide range of temperature and carrier density, with its characteristics sensitive to the band topology of graphene layers. These observations can be attributed to a new type of quantum interference between drag processes, with the interference pathway comprising different carrier diffusion paths in the two constituent conductors. The emergence of such effect relies on the formation of superimposing planar diffusion paths, among which the impurity potentials from intermediate insulating spacer play an essential role. Our findings establish an ideal platform where the interplay between quantum interference and many-body interaction is essential.

reciprocal space. Here, the paired superimposing planar paths ( a α and p α , a β and p β ) belong to the same drag process. This differs from the e-e case, in which the paired paths belong to different drag processes.

Supplementary Table 2 | Phase analyses in the e-h/h-e region
note: Cases A-F are schematically shown in Supplementary Fig. 8.  Supplementary Fig. 8.

Supplementary Note 1. Validity of drag measurements
To verify the validity of the drag measurements, we first conducted comparative experiments by interchanging the active and passive layers (two opposite setups shown in Supplementary Fig.   2a). As demonstrated in Supplementary Figs. 2b-d, the as-obtained drag curves using these two drag set-ups are nearly identical, well obeying the expected Onsager reciprocity relation.
We futher checked the Vp -Ia relationship at different temperatures using setup 1. A good liner Vp -Ia relation occurred at temperatures above 100 K ( Supplementary Fig. 3). Lowering the temperature leads to unexpected fluctuations, hindering the extraction of accurate Rdrag via linear fitting. Similar fluctuations have been observed previously, and were interpreted as mesoscopic drag fluctuations as a consequence of phase coherent quantum transport within the constituent layers [1][2][3].

Supplementary Note 2. Theoretical analysis on the high field B 2 dependence of drag resistance
The high field behavior of B 2 dependence for the drag resistance has been widely reported in previous experimental studies [4][5][6]. Below, we would like to give an explanation for such B 2 dependence based on the Drude-like model for Coulomb drag.
According to previous study [7], for a drag system wherein the two constituent graphene layers are identical, drag resistivity is given by Here = e − h and = e + h are the charge and quasiparticle densities in each layer, r0 is the zero-field residual resistance of graphene at the charge neutral point, R0 and RH are the Drude and Hall resistances far from the charge neutral point, 0 D is the zero-field drag resistance far from the charge neutral point, and ̅ ± is defined as where W is sample width, is the mean quasiparticle kinetic energy, ph −1 and Q −1 are the quasiparticle relaxation rate due to the electron-phonon scattering and the inter-layer Coulomb interaction, respectively.
For a sample with small W ( ≪ ± ), one will find ̅ ± ≈ 2 /(3 ± 2 ) and then we have Here B is the magnetic field and μ is the carrier mobility. We can draw two conclusions from this expression: (i) The zero field drag resistivity, given by −

Supplementary Note 3. Repeatability of magneto-drag behaviors in BLG/BLG devices
Supplementary Fig. 4 presents the evolution of magneto-drag behaviors with varying temperatures in the e-h region. It is clearly seen that the low-field correction gets suppressed with increasing temperature (Supplementary Fig. 4b), consisting with the evolution behavior observed in the e-e region (see Fig. 2b in the main text).
In Supplementary Fig. 5 we further plotted the MRdrag data measured at different carrier densities for the h-h, e-h and h-e regions. All the MRdrag curves show a clear low-field deviation, with the magnitude monotonically weakening as the carrier densities of BLG layers increase.
These observations also consist well with the results from the e-e region (Fig. 2c in the main text).
To further evaluate the repeatability of the magneto-drag behaviors, we conducted systematic measurements on another BLG/BLG device. Supplementary Fig. 6a shows Rdrag as functions of nT and nB obtained at 200 K, from which four distinct regions are clearly seen. Low-field deviations are evident for the MRdrag curves taken in both the e-e region ( Supplementary Fig. 6b) and the e-h region ( Supplementary Fig. 6c), and show identical characteristics with those obtained in the device demonstrated in the main text (Fig. 2c in the main text and Supplementary Fig. 5c).

A. Probability amplitude of the drag process
According to the leading-order Feynman diagram of drag conductivity ( Supplementary Fig.   7a) [8,9], a typical drag process can be described as follows ( Supplementary Fig. 7b): 1) A carrier in the active layer ("active carrier") starts at 0 a and a carrier in the passive layer ("passive carrier") at 0 p ; 2) These two carriers interact Coulombically when they travel separately to 1 a and 1 p along the diffusion paths 1 and 3 ; 3) After the Coulomb scattering, they end up at 2 a and 2 p after traveling along paths 2 and 4 , respectively. The probability amplitude of such an inter-layer transport process is given by: Here, describes the inter-layer Coulomb interaction and is the propagation amplitude along diffusion path . We note that the inter-layer Coulomb interaction will transfer energy ∆ from the active layer to the passive layer, and thus introduce phase factors i∆ and −i∆ to the propagating amplitudes of the active carrier and the passive carrier, respectively. However, these two phase factors are involved in 2 and 4 , respectively, and will cancel out with each other eventually. That is, the inter-layer Coulomb interaction does not lead to any dephasing for the overall drag process.
Then the probability ( 2 p , 0 a ) for an active carrier at 0 a inducing a passive carrier moving to 2 p is: Here, 2 a and 0 p are integrated since they are not recorded in the drag measurement. As the drag conductivity can be obtained from ( 2 p , 0 a ) and the knowledge of current vertices in Supplementary Fig. 7a, analysis of ( 2 p , 0 a ) can provide a qualitative understanding of the drag conductivity/current.

B. Inter-layer interference between drag processes
For a pair of drag processes "α" and "β" (see Supplementary Fig. 7c), the interference term in ( 2 p , 0 a ) is given by 2| α β |cos( α − β ) . Here, and represent the probability amplitude of the drag process and the corresponding phase, respectively. Note that these two drag processes α and β have the same initial and final states, i.e., { 0 a , 0 p } and { 2 a , 2 p }, which is necessary for the emergence of interference. As a result, the diffusion paths involved in the interference, i.e., { 1 α , 2 α , 1 β , 2 β } and { 3 α , 4 α , 3 β , 4 β }, form a closed loop within both the active and passive layers (Supplementary Fig. 7c). The phase of the probability amplitude for drag process α or β can be obtained by adding up the phases of constituent diffusion paths: In general, interferences of nearly all pairs of drag processes cancel out after being averaged over all possible diffusion paths. Only pairs of drag processes with constant α − β that are independent of constituent paths will have observable interference effect that contributes to the drag signal. To identify these paired drag processes, we first divide the corresponding eight diffusion paths, i.e., { α , β | = 1, 2, 3, 4}, into four groups of two. The two paths in each group are required to have a definite phase relation and contribute to a constant in α − β . Here, a definite phase relation refers to either a constant difference or a sign reversal, depending on that these two paths belong to different drag processes or the same one. For inter-layer QI, these two paths are further required to reside respectively in the two separate layers. (

1) e-e/h-h region
For the e-e/h-h region, the propagation amplitudes along the two superimposing planar paths have the same phase, that is: Here, we note that time reversal does not change the propagation amplitude. Supplementary Table   1 lists the calculated phase difference of two drag processes α − β , from which only the scenarios shown in Supplementary Figs. 8a,f can survive the impurity average since the overall phase differences for these circumstances are constant and therefore independent of constituent diffusion paths.

(2) e-h/h-e region
For the e-h/h-e region, the propagation amplitudes along the two superimposing planar paths have opposite phases (see Supplementary Note 4E), that is: After conducting analyses similar to the e-e/h-h case, we conclude that only the scenarios shown in Supplementary Figs. 8b,e survive the impurity average (Supplementary Table 2).

C. Magnetic-field-induced phase difference
When a vertical magnetic field is applied, an additional phase ( = ±∫ • ) arises for a carrier propagating along path under a vector potential , with its sign depending on the polarity and the propagation direction of the carrier. For the above identified pairs of drag processes, the corresponding can be readily obtained, as listed in Supplementary Table 3 and Table 4. It is concluded that only the interference shown in Supplementary Fig. 8a (Fig. 8b) for the e-e/h-h region (e-h/h-e region) is destroyed by the magnetic field, and responsible for the observed lowfield corrections of drag resistance.

D. Quantum correction to the drag resistance
The finally established inter-layer QI for the e-e and e-h regions are schematically shown in Fig. 3b (right panel) in the main text and Supplementary Fig. 9a, respectively. Each pair of superimposing planar paths in these interferences are interrelated by inter-layer mirror reflection and time reversal. For convenience, we define a α = { 1 α , 2 α } and p α = { 3 α , 4 α }. As detailed in the main text, in the reciprocal space, the drag process involved in the inter-layer QI corresponds to the case that an active carrier with momentum scatters a passive carrier into the − state ( Fig.   3c in the main text and Supplementary Fig. 9b), which can be referred to as inter-layer backscattering.
Assuming that the drive current is a = a a ( ), the correction induced by the above interlayer QI to the drag current would be p cor = p p (− ), with its direction opposite to a for both the e-e and e-h regions. Here, and are the charge and velocity of carrier respectively. We note that in our model, a hole with momentum means that the state is unoccupied. Thus, the hole with momentum − in the conduction band has the same velocity as the electron with momentum in valence band.
Similar to the analysis for the e-e region presented in the main text, here we discuss the correction of inter-layer QI to the drag resistance in the e-h region. For typical Coulomb drag obeying momentum transfer mechanism, the classical drag current p cla will flow along the opposite direction as the drive current a in the e-h region. The enhancement of inter-layer backscattering due to QI will leads to an increase in the total drag current (Ip cla + Ip cor ). The corresponding increase in accumulated open-circuit voltage between the electrodes (i.e., the measured Vp) leads to an increase in the magnitude of drag resistance. This is consistent with the experimentally obtained deviation of zero-field Rdrag (Fig. 1f in the main text).
As detailed in Supplementary Note 4C, applying a magnetic field will break the interference.
Thus, the quantum correction to the drag current p cor will be suppressed with increasing magnetic field, leading to negative MRdrag for the e-h region at low-field regime. This conclusion also consists well with our experimental data ( Supplementary Figs. 4b and 5c).

E. Propagation amplitudes of carriers
To investigate the impurity scatterings of a carrier in a crystal, the -matrix in scattering theory is defined as: Here, I is the impurity potential, 0 is the Hamiltonian of the perfect crystal, and is the energy of carrier.
For convenience, we will discuss the propagation amplitude in the reciprocal (quasimomentum) space, and the conclusions hold for the real space as well. A diffusion path in the reciprocal space can be defined as a sequence: where and ′ are the initial and final momenta, respectively, and are intermediate momenta.
For monolayer and bilayer graphene, we will consider the commonly used low-energy two band models [10][11][12] with band index = ±1 corresponding to conduction and valence bands, respectively. In these models, the matrix elements of the impurity potential are independent of band index: ⟨ ′ | I | ⟩ = ⟨− ′ | I |− ⟩ = ′ I [11]. For an electron in the conduction band with = +1, the propagation amplitude along is: Here, 1 , 2 ≫ d . The classical drag resistivity is d = − d 1 2 , which is a direct measurement of the inter-layer Coulomb interaction [14].

C. Inter-layer quantum correction
For the inter-layer QI effect, the conductivity matrix can be rewritten as: .